Lie Group Formulation of Recursive Dynamics Algorithms of Higher Order for Floating-Base Robots

TL;DR

This paper develops higher-order recursive dynamics algorithms for floating-base robots using Lie group theory, providing closed-form equations and demonstrating practical benefits through a 12-DoF aerial manipulator example.

Contribution

It introduces a Lie group formulation for higher-order derivatives of robot dynamics, including closed-form equations and passivity properties, with applications to complex robotic systems.

Findings

Algorithms compute derivatives up to fifth order efficiently.

The articulated inertia tensor remains constant across derivatives.

Computational cost scales quadratically with derivative order, outperforming automatic differentiation.

Abstract

In this paper, we describe procedures for computing higher-order time derivatives of the Lie-group Newton-Euler, Articulated-Body Inertia, and hybrid dynamics algorithms for floating-base trees, where the base configuration evolves on SE(3) and the attached mechanism is an open kinematic tree with configuration on the (n1+n2)-dimensional manifold T^{n1} \times R^{n2}, using spatial representation of twists. After presenting the algorithms, we collect the resulting recursions into closed-form equations of motion, identifying an admissible Coriolis matrix satisfying the passivity property, and showing that the articulated inertia tensor remains unchanged across all time derivatives. We then apply the developed methods to a 12-DoF aerial manipulator to derive analytical expressions for its geometric forward and inverse dynamics along with their first time derivatives whereas the numerical simulations successfully evaluate these dynamics up to fifth order. Finally, to demonstrate their practical utility, we benchmark the proposed extensions and show that, in the considered tests, their computational cost scales quadratically with the derivative order, whereas the automatic-differentiation baseline exhibits exponential scaling.